Follow the Money

Today, a different kind of analyzed bracket.

Most of the brackets posted here have shown the prospects of the player occupying a certain line in terms of the chance of winning the tournament as a whole from that line. In this new bracket, the outlook is couched in terms of expected prize money – the player’s aggregate chance of winning any of six prizes.

The bracket I’ve analyzed is one of the possible brackets for the main even at this weekend’s Viking Classic backgammon tournament in Minnesota. The analysis gives me a chance to show a few new things besides the money, also.

The analyzed brackets are here: OdinUpper and OdniLower. The analysis by player skill is in a table below rather than tucked into a corner of the upper bracket, as has been its wont.

The tournament is being run by April Kennedy and Michael Mesich, a pair of extraordinarily energetic directors. Their brackets will have none but good drops and properly spread byes. I know this because I drew the lines and placed the drops. But the brackets have become works of art – a look even if you don’t give a hang about how they’re going to play. There’s an example on the tourney’s Facebook page.

The event is the “Odin Open”, the upper division of the tourney’s main event. I’ve simulated the tourney assuming that it will go off with 32 players, which would be ideal, from a fairness perspective.

It’s being run as a consolation event rather than a full double elimination – the lower bracket becomes it’s own tournament, with 40% of the prize fund paid to four consolation places. With two payouts in the upper bracket, there are a total of six money winners. That’s a deeper payout than most backgammon tourneys, but there’s added money in the main event (and several others), so the payouts are still substantial.

The skill/luck balance is set at 1:3, as discussed in Skill and Luck in Backgammon, and All that Luck. The results show that chance is a major factor – the best player wins less than 11% of the time, and is in the money less than 42%. Even the weakest player stands to win back, on average, a little over half the $300 entry fee.

I’ve set the elite threshold at zero, which means that only above-average players are assuming to be playing in the upper division. This is particularly appropriate for Minnesota, where all the children are above average. The table below, which reports the expectation by skill level rather than bracket line, shows that, but for the added money, a player would have to have a Z-score greater than one to have a positive expectation.

This is yet another example of how the few at the top exploit the rest. Now that they have risen en masse and elected one of their own to be president of the United States, we should, perhaps, refer to the great mass of ordinary people in formal statistical terminology: as sub-standard deviants.

The prize pool is divided 40/20/20/10/5/5, for the upper winner, the upper runner-up, the consolation winner, the consolation runner-up, and the two consolation semi-finalists. The thing to note is that it takes, on average, considerable more skill to win the 20% for winning the consolation than it is to win the 20% for runner-up. Perhaps the consolation winner should have a larger share. I intend to use data like this in order to derive the new fairness (D) measure discussed in an earlier post. But equating the runner-up and the consolation champ is common, perhaps because it seems unfair (in a fairness (A) sense) to pay anyone in the lower bracket more than anyone in the upper.

There are other fairness (A) notions that seem to attach to setting a payout schedule. Another, for example, which is comfortably observed in this case, is that there should be no payout less than an entry fee.

Once a fairness (D) measure has been designed and tested, it may be that payout schedules can be optomized in much the same way that bracket designs are now.

Here is the result table for players. The bold numbers in the second row are the mean skill level associated with each prize.